Numerical integration errors and volumetric locking in the near-incompressible limit are two outstanding issues in Galerkin-based meshfree computations. In this paper, we present a modified Gaussian integration scheme on background cells for meshfree methods that alleviates errors in numerical integration and ensures patch test satisfaction to machine precision. Secondly, a lockingfree small-strain elasticity formulation for meshfree methods is proposed, which draws on developments in assumed strain methods and nodal integration techniques. In this study, maximum-entropy basis functions are used; however, the generality of our approach permits the use of any meshfree approximation. Various benchmark problems in two-dimensional compressible and near-incompressible small strain elasticity are presented to demonstrate the accuracy and optimal convergence in the energy norm of the maximumentropy meshfree formulation.
A novel maximum-entropy meshfree method that we recently introduced in Ref.[1] is extended to Stokes flow in two dimensions and to three-dimensional incompressible linear elasticity. The numerical procedure is aimed to remedy two outstanding issues in meshfree methods: the development of an optimal and stable formulation for incompressible media, and an accurate cell-based numerical integration scheme to compute the weak form integrals.On using the incompressibility constraint of the standard u-p formulation, a u-based formulation is devised by nodally averaging the hydrostatic pressure around the nodes. A modified Gauss quadrature scheme is employed, which results in a correction to the stiffness matrix that alleviates integration errors in meshfree methods, and satisfies the patch test to machine accuracy.The robustness and versatility of the maximum-entropy meshfree method is demonstrated in three-dimensional computations using tetrahedral background meshes for integration. The meshfree formulation delivers optimal * Corresponding author Email address: alortiz@ucdavis.edu (A. Ortiz) December 25, 2010 rates of convergence in the energy and L 2 -norms. Inf-sup tests are presented to demonstrate the stability of the maximum-entropy meshfree formulation for incompressible media problems.
Preprint submitted to Elsevier
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